On inverse problem with phase retrieval for an inclined plane in the parabolic approximation

Abstract

The inverse problem of amplitude reconstruction on an inclined plane based on the values of amplitude or its module as recorded on semi-infinite plane orthogonal to the beam propagation direction is considered within the framework of 2D parabolic equation. It is demonstrated that this inverse problem, in case when the complex image plane amplitude is known, can be reduced to a singular Cauchy-type integral equation. The existence of its solutions requires that certain conditions be met but if a solution exists it is necessary unique. The obtained integral equation is then approximated piece-wisely and the resulting linear algebraic system is solved numerically while applying necessary regularization procedures to enhance the stability of its solutions. Finally, an iterative method of phase retrieval is developed and a set of numerical experiments is performed.

Keywords:

• Inverse problem
• integral equation
• iterative algorithm
• computational imaging

PACS CLASSIFICATIONS:

• 61.10.Nz
• 41.50.+h
• 02.30.Rz

Acknowledgments

The work was supported by a New Scientific Groups Project of P.N. Lebedev Physical Institute: Methods of optical photons storage and of the output X-ray beam collimation of a laser-electron source for cardiology.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by P.N. Lebedev Physical Institute [15].